Mark DominusI wonder if perhaps arithmetic division was invented before multiplication.
Because people would have had goods in need of fair division long before they would have had equal groups of already-divided goods that needed to be totaled.
luksfarris@mjd maybe multiplication came from an early desire to forecast yield. Two hunters grabbing three animals each, etc
@luksfarris Conceivably, but I don't find your example compelling. Early people can count to six on their fingers, they don't need to know that 2×3=6.
If it were 7 people going out to maybe get 6 squirrels each, why would they care to know it was 42 squirrels rather than just a big heap of squirrels?
@mjd didn't mean to refute your original point, I agree division comes first. I suppose multiplication could make more sense for agriculture than for foraging
@luksfarris I didn't think you were trying to refute me, I thought you were contributing to the conversation. I hope my own contribution didn't seem brusque! I was just trying to continue as we had begun.
Your observation about agriculture makes a lot of sense to me. Thinking in terms of N fields each producing a certain number of sacks of grain, multiplication makes a lot of sense!
Your observation about agriculture makes a lot of sense to me. Thinking in terms of N fields each producing a certain number of sacks of grain, multiplication makes a lot of sense!
@luksfarris And your agriculture idea brings out something I was struggling to articulate: why would someone need to know how many sacks of grain to expect? It's a large number of no use _unless_ they were planning to divide it again.
For example they might compute “do we have enough to survive the winter” by computing how many sacks could be allotted to each family.
Or “A granary can hold 80 sacks. Do we need to dig more granaries?”
But why multiply in the first place, if not to divide again later?
@mjd I agree, that makes a lot of sense!!!
novalis@mjd @luksfarris I think of multiplication as coming from a generalization of skip-counting. Like, you're trying to inventory a bunch of stuff, so instead of counting, one amphora, two amphorae, etc, you count two, four, six, eight, ten makes a boxful, one box, two boxes, etc.
You need formalized multiplication when you have enough (fungible) stuff that counting becomes hard.
You get division in a similar way -- if you have those 42 squirrels and six people, you deal them out into six piles one-at-a-time (keeping track of where you started in case of remainders). But I think most ways of formalizing it rely on already having multiplication (at least doubling).